Bistable flows in precessing spheroids

Precession driven flows are found in any rotating container filled with liquid, when the rotation axis itself rotates about a secondary axis that is fixed in an inertial frame of reference. Because of its relevance for planetary fluid layers, many works consider spheroidal containers, where the uniform vorticity component of the bulk flow is reliably given by the well-known equations obtained by Busse in 1968. So far however, no analytical result on the solutions is available. Moreover, the cases where multiple flows can coexist have not been investigated in details since their discovery by Noir et al. (2003). In this work, we aim at deriving analytical results on the solutions, aiming in particular at, first estimating the ranges of parameters where multiple solutions exist, and second studying quantitatively their stability. Using the models recently proposed by Noir \& C{\'e}bron (2013), which are more generic in the inviscid limit than the equations of Busse, we analytically describe these solutions, their conditions of existence, and their stability in a systematic manner. We then successfully compare these analytical results with the theory of Busse (1968). Dynamical model equations are finally proposed to investigate the stability of the solutions, which allows to describe the bifurcation of the unstable flow solution. We also report for the first time the possibility that time-dependent multiple flows can coexist in precessing triaxial ellipsoids. Numerical integrations of the algebraic and differential equations have been efficiently performed with the dedicated script FLIPPER (supplementary material)

Precession-driven flows in non-axisymmetric ellipsoids

We study the flow forced by precession in rigid non-axisymmetric ellipsoidal containers. To do so, we revisit the inviscid and viscous analytical models that have been previously developed for the spheroidal geometry by, respectively, Poincaré (Bull. Astronomique, vol. XXVIII, 1910, pp. 1-36) and Busse (J. Fluid Mech., vol. 33, 1968, pp. 739-751), and we report the first numerical simulations of flows in such a geometry. In strong contrast with axisymmetric spheroids, where the forced flow is systematically stationary in the precessing frame, we show that the forced flow is unsteady and periodic. Comparisons of the numerical simulations with the proposed theoretical model show excellent agreement for both axisymmetric and non-axisymmetric containers. Finally, since the studied configuration corresponds to a tidally locked celestial body such as the Earth's Moon, we use our model to investigate the challenging but planetary-relevant limit of very small Ekman numbers and the particular case of our Moo

Libration driven elliptical instability

The elliptical instability is a generic instability which takes place in any rotating flow whose streamlines are elliptically deformed. Up to now, it has been widely studied in the case of a constant, non-zero differential rotation between the fluid and the elliptical distortion with applications in turbulence, aeronautics, planetology and astrophysics. In this letter, we extend previous analytical studies and report the first numerical and experimental evidence that elliptical instability can also be driven by libration, i.e. periodic oscillations of the differential rotation between the fluid and the elliptical distortion, with a zero mean value. Our results suggest that intermittent, space-filling turbulence due to this instability can exist in the liquid cores and sub-surface oceans of so-called synchronized planets and moons

Libration driven multipolar instabilities

We consider rotating flows in non-axisymmetric enclosures that are driven by libration, i.e. by a small periodic modulation of the rotation rate. Thanks to its simplicity, this model is relevant to various contexts, from industrial containers (with small oscillations of the rotation rate) to fluid layers of terrestial planets (with length-of-day variations). Assuming a multipolar $n$-fold boundary deformation, we first obtain the two-dimensional basic flow. We then perform a short-wavelength local stability analysis of the basic flow, showing that an instability may occur in three dimensions. We christen it the Libration Driven Multipolar Instability (LDMI). The growth rates of the LDMI are computed by a Floquet analysis in a systematic way, and compared to analytical expressions obtained by perturbation methods. We then focus on the simplest geometry allowing the LDMI, a librating deformed cylinder. To take into account viscous and confinement effects, we perform a global stability analysis, which shows that the LDMI results from a parametric resonance of inertial modes. Performing numerical simulations of this librating cylinder, we confirm that the basic flow is indeed established and report the first numerical evidence of the LDMI. Numerical results, in excellent agreement with the stability results, are used to explore the non-linear regime of the instability (amplitude and viscous dissipation of the driven flow). We finally provide an example of LDMI in a deformed spherical container to show that the instability mechanism is generic. Our results show that the previously studied libration driven elliptical instability simply corresponds to the particular case $n=2$ of a wider class of instabilities. Summarizing, this work shows that any oscillating non-axisymmetric container in rotation may excite intermittent, space-filling LDMI flows, and this instability should thus be easy to observe experimentally

Libration-driven multipolar instabilities

We consider rotating flows in non-axisymmetric enclosures that are driven by libration, i.e. by a small periodic modulation of the rotation rate. Thanks to its simplicity, this model is relevant to various contexts, from industrial containers (with small oscillations of the rotation rate) to fluid layers of terrestrial planets (with length-of-day variations). Assuming a multipolar $n$ -fold boundary deformation, we first obtain the two-dimensional basic flow. We then perform a short-wavelength local stability analysis of the basic flow, showing that an instability may occur in three dimensions. We christen it the libration-driven multipolar instability (LDMI). The growth rates of the LDMI are computed by a Floquet analysis in a systematic way, and compared to analytical expressions obtained by perturbation methods. We then focus on the simplest geometry allowing the LDMI, a librating deformed cylinder. To take into account viscous and confinement effects, we perform a global stability analysis, which shows that the LDMI results from a parametric resonance of inertial modes. Performing numerical simulations of this librating cylinder, we confirm that the basic flow is indeed established and report the first numerical evidence of the LDMI. Numerical results, in excellent agreement with the stability results, are used to explore the nonlinear regime of the instability (amplitude and viscous dissipation of the driven flow). We finally provide an example of LDMI in a deformed spherical container to show that the instability mechanism is generic. Our results show that the previously studied libration-driven elliptical instability simply corresponds to the particular case $n= 2$ of a wider class of instabilities. Summarizing, this work shows that any oscillating non-axisymmetric container in rotation may excite intermittent, space-filling LDMI flows, and this instability should thus be easy to observe experimentall

Earth rotation prevents exact solid-body rotation of fluids in the laboratory

International audience–We report direct evidence of a secondary flow excited by the Earth rotation in a water-filled spherical container spinning at constant rotation rate. This so-called tilt-over flow essentially consists in a rotation around an axis which is slightly tilted with respect to the rotation axis of the sphere. In the astrophysical context, it corresponds to the flow in the liquid cores of planets forced by precession of the planet rotation axis, and it has been proposed to contribute to the generation of planetary magnetic fields. We detect this weak secondary flow using a particle image velocimetry system mounted in the rotating frame. This secondary flow consists in a weak rotation, thousand times smaller than the sphere rotation, around a horizontal axis which is stationary in the laboratory frame. Its amplitude and orientation are in quantitative agreement with the theory of the tilt-over flow excited by precession. These results show that setting a fluid in a perfect solid body rotation in a laboratory experiment is impossible — unless tilting the rotation axis of the experiment parallel to the Earth rotation axis. Introduction. – There are few examples of fluid mechanics experiments at the laboratory scale in which the Earth's Coriolis force has a measurable influence. Such experiments may be considered as fluid analogues to the Foucault pendulum. The most popular instance is certainly the drain of a bathtube vortex [1]. Although this is the subject of common misconception, it is actually possible to detect the influence of the Earth's rotation on the vortex, but only under extremely careful experimental conditions, far from the everyday experience [2]. Thermal convection is another example, in which a slow drift of the large-scale flow due to the Earth rotation has been detected in very controlled systems [3, 4]. In this letter we describe an experiment which may be considered as the most simple fluid Foucault pendulum: it consists in a volume of water enclosed in a spherical container spinning at constant rotation rate Ω 0 (fig. 1). After a transient known as spin-up, the water is expected to rotate as a solid body at the same rate Ω 0 [5]. The (a

Shallow water waves generated by subaerial solid landslides

Subaerial landslides are common events, which may generate very large water waves. The numerical modelling and simulation of these events are thus of primary interest for forecasting and mitigation of tsunami disasters. In this paper, we aim at describing these extreme events using a simplified shallow water model to derive relevant scaling laws. To cope with the problem, two different numerical codes are employed: one, SPHysics, is based on a Lagrangian meshless approach to accurately describe the impact stage whereas the other, Gerris, based on a two-phase finite-volume method is used to study the propagation of the wave. To validate Gerris for this very particular problem, two numerical cases of the literature are reproduced: a vertical sinking box and a 2-D wedge sliding down a slope. Then, to get insights into the problem of subaerial landslide-generated tsunamis and to further validate the codes for this case of landslides, a series of experiments is conducted in a water wave tank and successfully compared with the results of both codes. Based on a simplified approach, we derive different scaling laws in excellent agreement with the experiments and numerical simulation

Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits

We consider the hydrodynamic stability of homogeneous, incompressible and rotating ellipsoidal fluid masses. The latter are the simplest models of fluid celestial bodies with internal rotation and subjected to tidal forces. The classical problem is the stability of Roche–Riemann ellipsoids moving on circular Kepler orbits. However, previous stability studies have to be reassessed. Indeed, they only consider global perturbations of large wavelength or local perturbations of short wavelength. Moreover many planets and stars undergo orbital motions on eccentric Kepler orbits, implying time-dependent ellipsoidal semi-axes. This time dependence has never been taken into account in hydrodynamic stability studies. In this work we overcome these stringent assumptions. We extend the hydrodynamic stability analysis of rotating ellipsoids to the case of eccentric orbits. We have developed two open-source and versatile numerical codes to perform global and local inviscid stability analyses. They give sufficient conditions for instability. The global method, based on an exact and closed Galerkin basis, handles rigorously global ellipsoidal perturbations of unprecedented complexity. Tidally driven and libration-driven elliptical instabilities are first recovered and unified within a single framework. Then we show that new global fluid instabilities can be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits. Their existence is confirmed by a local analysis and direct numerical simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital eccentricity may have a strong destabilising effect in celestial fluid bodies, which may lead to space-filling turbulence in most of the parameters range

Tidal instability in a rotating and differentially heated ellipsoidal shell

The stability of a rotating flow in a triaxial ellipsoidal shell with an imposed temperature difference between inner and outer boundaries is studied numerically. We demonstrate that (i) a stable temperature field encourages the tidal instability, (ii) the tidal instability can grow on a convective flow, which confirms its relevance to geo- and astrophysical contexts and (iii) its growth rate decreases when the intensity of convection increases. Simple scaling laws characterizing the evolution of the heat flux based on a competition between viscous and thermal boundary layers are derived analytically and verified numerically. Our results confirm that thermal and tidal effects have to be simultaneously taken into account when studying geophysical and astrophysical flows